HOCHSCHILD COHOMOLOGY AND LINCKELMANN COHOMOLOGY FOR BLOCKS OF FINITE GROUPS JONATHAN PAKIANATHAN AND SARAH WITHERSPOON Abstract. Let G be a finite group, F an algebraically closed field of finite characteristic p, and let B be a block of FG. We show that the Hochschild and Linckelmann cohomology rings of B are isomorphic, modulo their radicals, in the cases where (1) B is cyclic and (2) B is arbitrary and G either a nilpotent group or a Frobenius group (p odd). (The second case is a consequence of a more general result). We give some related results in the more general case that B has a Sylow p-subgroup P as a defect group, giving a precise local description of a quotient of the Hochschild cohomology ring. In case P is elementary abelian, this quotient is isomorphic to the Linckelmann cohomology ring of B, modulo radicals. 1991 Mathematics Subject Classification. Primary: 16E40, 20J06 1. Introduction Let G be a finite group and F an algebraically closed field of posi- tive characteristic p dividing the order of G. Let B be a block of the group algebra FG, that is an indecomposable ideal direct summand of FG. In [14, 15], Linckelmann defines the cohomology ring LH∗(B) (our notation) of the block B of FG to be a subring of certain stable elements in the group cohomology ring H∗(P, F), where P is a defect group of B. (See Definition 2.1.) Linckelmann then defines an injective ring homomorphism γ from the block cohomology ring LH∗(B) to the Hochschild cohomology ring HH∗(B) of B [14]. We are interested in a better understanding of the map γ connecting these two cohomology rings. As HH0(B) generally has dimension over F larger than one, γ is not in general an isomorphism. However, if we take the quotient of each ring by its (Jacobson) radical, we still have an injective ring homomorphism, which is now an isomorphism in degree 0. One is now led to the following question: Date: February 22, 2002. Research of the second author supported in part by National Science Foundation Grant #DMS-9970119 and National Security Agency Grant #MDA904-01-1-0067. 1 2 JONATHAN PAKIANATHANAND SARAH WITHERSPOON When does Linckelmann’s injection γ : LH∗(B) → HH∗(B) induce an isomorphism γ modulo radicals? We point out that as these cohomology rings are finitely generated graded commutative rings, we need only check that such an isomor- phism exists, and Linckelmann’s injection γ will then automatically induce an isomorphism γ. It is known that LH∗(B) and HH∗(B) have the same Krull dimension, that of H∗(P, F), the rank of P ([15, Corol- lary 4.3(ii)] or [11, Theorem 4.4]). If B = B0 is the principal block, γ is known to be an isomorphism in the cases where G is a p-group, G is abelian, and a few other specific cases [21, Sections 10 and 11], as well as the case where B0 is cyclic, that is its defect groups are cyclic [22, Theorem 3]. In §3, we extend these results to prove: Theorem 3.1. Let G be a group with normal Sylow p-subgroup P such that for any k ∈ PCG(P ) − 1, CG(k) ≤ PCG(P ). (For example, G a nilpotent group or G a Frobenius group where p divides the order of the Frobenius kernel.) Then for any block B of FG, we have that LH∗(B) and HH∗(B) are isomorphic modulo their radicals. As a consequence, we give an affirmative answer to the question for all cyclic blocks: Corollary 3.5. Let G be any finite group, and B any block of FG having a cyclic defect group. Then the Linckelmann cohomology ring LH∗(B) is isomorphic to the Hochschild cohomology ring HH∗(B), mod- ulo radicals. We then give further examples: An analogous result is true for the principal blocks of A5 and SL2(8). These examples use Theorem 3.1 and Brou´e’s abelian defect conjecture, which is known to hold for these groups by work of Rickard and Rouquier. They also suggest a strategy for handling a larger class of examples. In §4, we give some related results in case P is a Sylow p-subgroup of G (now not necessarily normal), and B any block of G with defect group ∗ P (e.g. the principal block). We study the quotient ring HHP (B) of the Hochschild cohomology ring of B modulo the ideal of proper transfers (see Definition 4.1) after proving some general module-theoretic results HOCHSCHILD AND LINCKELMANN COHOMOLOGY 3 about this cohomology quotient. We give the structure of this quotient ∗ HHP (B) in terms of local information: Theorem 4.2. Let B be a block of FG with defect group the Sylow p-subgroup P of G. Let K = PCG(P ), and b a block of FK such that B is the unique block covering b. Then ∗ ∼ ∗ F F NG(b) HHP (B) = (HP (P, ) ⊗ Z(P )) , where FZ(P ) is the group algebra of the center of P . ∗ In particular, when P is elementary abelian, this quotient HHP (B) is isomorphic, modulo radicals, to Linckelmann cohomology (Corollary 4.10). We give further examples of blocks of symmetric groups of de- fect 2 (p odd). In this case, Hochschild cohomology and Linckelmann cohomology are again isomorphic, modulo their radicals. These ex- amples use Theorem 3.1 and Chuang’s proof of Brou´e’s abelian defect conjecture for these blocks. A possible application of our work, particularly if it may be extended to include larger classes of groups and/or blocks, is to the study of vari- eties for blocks. In [15], Linckelmann develops such a theory, where the variety associated to a block B is the maximal ideal spectrum of the block cohomology ring LH∗(B). Some unpublished work of Siegel [20] also gives a theory of varieties for blocks, where this time the variety associated to a block is the maximal ideal spectrum of its Hochschild cohomology ring HH∗(B). In cases where Linckelmann’s block coho- mology and the Hochschild cohomology of the block are isomorphic modulo their radicals (or more generally F-isomorphic), these two va- rieties associated to the block will be the same, and so both theories may potentially be exploited to obtain further information. We would like to thank the referee for many useful comments and suggestions. 2. Preliminary remarks We will use a number of results from [3] on subpairs and their partial order. Let B be any block of FG. Let (P, BP ) be a Sylow B-subpair of G, unique up to conjugacy. In particular, P is a defect group of B and BP is a block of FCG(P ). If R is any subgroup of P , then there exists a unique block BR of FCG(R) such that (R, BR) ≤ (P, BP ). Let NG(BR) be the subgroup of NG(R) fixing BR setwise, under conjugation. 4 JONATHAN PAKIANATHANAND SARAH WITHERSPOON Definition 2.1 (Linckelmann). Let B be any block of FG with defect group P . The cohomology ring of the block B of G is the subring LH∗(B) of H∗(P, F) consisting of all [ζ] ∈ H∗(P, F) satisfying g P P resR([ζ]) = resR([ζ]) for any subgroup R of P , and any g ∈ NG(BR). Our definition is equivalent to that of Linckelmann: By [5, Theorem 1.8], BR is precisely the block of FCG(R) with block idempotent eR of [14, Definition 5.1]. (See also [15, p. 468].) We suppress Linckel- mann’s pointed group Pγ in our notation, as we do not explicitly use pointed groups in our definition, and in any case the definition (up to isomorphism) does not depend on the choice of Pγ . The next two remarks are due to Linckelmann [14, 15]. Remark 2.2. If B = B0 is the principal block, with defect group a Sylow p-subgroup P of G, it is easy to see that NG(BR) = NG(R) for all R ≤ P . Thus by the Alperin Fusion Theorem [1] and the standard description of stable elements [9, Corollary 4.2.7], it follows that ∗ ∗ LH (B0) = H (G, F). ∗ ∗ In this case, Linckelmann’s injection γ : LH (B0) → H (B0, B0) is the composition of the canonical injection H∗(G, F) ֒→ H∗(FG, FG) with ∗ ∗ the canonical projection H (FG, FG) ։ HH (B0). Remark 2.3. If the defect group P of B is abelian, then the inertial quotient E = NG(BP )/CG(P ) controls fusion [3, Proposition 4.2], and is in general a p′-group [8, Theorem 61.15]. Therefore ∗ ∗ ∗ LH (B) ∼= H (P, F)E ∼= H (P ⋊ E, F). (Here the superscript E denotes fixed points.) If a block b of NG(P ) is ∗ ∗ the Brauer correspondent of B, it follows that LH (B) ∼= LH (b), that is their block cohomology rings are isomorphic. It is not known whether their Hochschild cohomology rings HH∗(B) and HH∗(b) are isomorphic in this case. This would be a consequence of Brou´e’s abelian defect conjecture, that B and b are derived equivalent. Brou´e’s conjecture is known to hold in case P is cyclic [13, 16], as well as in a number of other cases. Let us look at an example so that the reader may see a sample of how Linckelmann’s injection induces an isomorphism modulo radicals. Example 2.4. Let G = S3, the symmetric group on three letters, and p = 2. The principal block B0 of FS3 is isomorphic to FC2 (where C2 HOCHSCHILD AND LINCKELMANN COHOMOLOGY 5 denotes a cyclic group of order 2).